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Chapter 1: Problem 67

If \(y\) is a linear function of \(x,\) then the ratio \(y / x\) is constant for all points on the graph at which \(x \neq 0\)

Short Answer

Expert verified
The ratio \( y/x \) is constant only if the function is of the form \( y = mx \) (i.e., \( c = 0 \)).

Step by step solution

01

Define a Linear Function

A linear function is typically defined as \[ y = mx + c \]where \( m \) is the slope and \( c \) is the y-intercept.
02

Express the Ratio y/x

For a linear function of the form \( y = mx + c \), express the ratio \( \frac{y}{x} \) as \[ \frac{y}{x} = \frac{mx + c}{x} \]. This simplifies to \[ \frac{y}{x} = m + \frac{c}{x} \].
03

Consider Cases where x 鈮� 0

To determine when \( \frac{y}{x} \) is constant, observe the expression \( m + \frac{c}{x} \). For the ratio to remain constant for all \( x eq 0 \), the term \( \frac{c}{x} \) should not change with \( x \), which occurs only if \( c = 0 \).
04

Conclusion on Linearity and Constant Ratio

If \( c = 0 \), then our function becomes \( y = mx \), and \( \frac{y}{x} = m \), which is indeed constant. For a general linear function \( y = mx + c \) with non-zero \( c \), \( \frac{y}{x} \) is not constant as depends on \( x \). Thus, for \( \frac{y}{x} \) to be constant for all points \( x eq 0 \), the linear function must be of the form \( y = mx \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
The slope of a linear function is a crucial concept to understand the graph of the function. Imagine the slope as a measure of how steep a line is on a graph. It tells us how much the value of \( y \) changes for every increase of 1 unit in \( x \). The formula for a linear function is given by \( y = mx + c \), where \( m \) represents the slope.
  • If \( m \) is positive, the line rises as it moves from left to right.
  • If \( m \) is negative, the line falls as it moves from left to right.
  • If \( m \) equals zero, the line is completely horizontal, indicating no change in \( y \) as \( x \) changes.
Grasping how the slope works will help you predict and understand the behavior of the linear function in different scenarios.
y-intercept
The y-intercept is another important element in the equation of a linear function \( y = mx + c \). The y-intercept, represented by \( c \), tells us where the line crosses the y-axis.
  • This intersection point occurs when \( x = 0 \).
  • The value of \( c \) gives the starting position of the line on the y-axis.
Understanding the concept of the y-intercept helps in visualizing the initial starting point and in graphing the linear equation accurately. If \( c = 0 \), the line will pass through the origin (0,0). This means our line equation is simply \( y = mx \), simplifying analysis regarding the line's behavior.
Constant Ratio
When dealing with linear functions, the term "constant ratio" refers to the condition where the ratio of \( y \) to \( x \), expressed as \( \frac{y}{x} \), remains the same across all points on the graph.
  • If \( c = 0 \), the function becomes \( y = mx \), making \( \frac{y}{x} = m \) a constant.
  • This constant nature implies that the relationship between \( y \) and \( x \) is directly proportional.
The constant ratio characteristic provides a method to predict \( y \) values for any given \( x \) value, as long as \( x eq 0 \). This helps greatly in real-world applications where such linear relationships are observed.
x not equal to zero
In the context of linear functions, the condition \( x eq 0 \) is critical for the analysis of the ratio \( \frac{y}{x} \). When \( x = 0 \), you potentially face undefined expressions in mathematics since you can鈥檛 divide by zero.
  • Thus, the condition \( x eq 0 \) prevents this problem, ensuring the expression \( \frac{y}{x} = m + \frac{c}{x} \) is well-defined.
  • This condition is crucial for verifying if the ratio of \( \frac{y}{x} \) remains constant.
In practical terms, always keeping \( x \) not equal to zero allows the examination and application of linear functions without mathematical conflict, making sure calculations remain valid wherever the function applies.

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Most popular questions from this chapter

Consider the function \(f(x)=\sin (1 / x)\) (a) Find a sequence of \(x\) -values that approach 0 such that \(\sin (1 / x)=0\) [Hint: Use the fact that \(\sin (\pi)=\sin (2 \pi)=\) \(\sin (3 \pi)=\ldots=\sin (n \pi)=0 .]\) (b) Find a sequence of \(x\) -values that approach 0 such that \(\sin (1 / x)=1\) IHint: Use the fact that \(\sin (n \pi / 2)=1\) if \(n=\) \(1,5,9, \dots .]\) (c) Find a sequence of \(x\) -values that approach 0 such that \(\sin (1 / x)=-1\) (d) Explain why your answers to any two of parts (a)-(c) show that \(\lim _{x \rightarrow 0} \sin (1 / x)\) does not exist.

Determine functions \(f\) and \(g\) such that \(h(x)=f(g(x)) .\) [Note: There is more than one correct answer. Do not choose \(f(x)=x \text { or } g(x)=x\).] $$h(x)=\sqrt{x^{2}+4}$$

Are the statements true or false? Give an explanation for your answer. The graph of \(f(x)=100\left(10^{x}\right)\) is a horizontal shift of the graph of \(g(x)=10^{x}\).

What does a calculator suggest about \(\lim _{x \rightarrow 0^{+}} x e^{1 / x} ?\) Does the limit appear to exist? Explain.

In Problems \(34-37\), is the function continuous for all \(x ?\) If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied. $$f(x)=\left\\{\begin{array}{ll} 2 x / x & x \neq 0 \\ 3 & x=0 \end{array}\right.$$
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